Patrick Forré

David Ruhe, Kaze Wong, Miles Cranmer et al. · cambridge

We propose parameterizing the population distribution of the gravitational wave population modeling framework (Hierarchical Bayesian Analysis) with a normalizing flow. We first demonstrate the merit of this method on illustrative experiments and then analyze four parameters of the latest LIGO/Virgo data release: primary mass, secondary mass, redshift, and effective spin. Our results show that despite the small and notoriously noisy dataset, the posterior predictive distributions (assuming a prior over the parameters of the flow) of the observed gravitational wave population recover structure that agrees with robust previous phenomenological modeling results while being less susceptible to biases introduced by less flexible models. Therefore, the method forms a promising flexible, reliable replacement for population inference distributions, even when data is highly noisy.

Benjamin Kurt Miller, Christoph Weniger, Patrick Forré

Likelihood-to-evidence ratio estimation is usually cast as either a binary (NRE-A) or a multiclass (NRE-B) classification task. In contrast to the binary classification framework, the current formulation of the multiclass version has an intrinsic and unknown bias term, making otherwise informative diagnostics unreliable. We propose a multiclass framework free from the bias inherent to NRE-B at optimum, leaving us in the position to run diagnostics that practitioners depend on. It also recovers NRE-A in one corner case and NRE-B in the limiting case. For fair comparison, we benchmark the behavior of all algorithms in both familiar and novel training regimes: when jointly drawn data is unlimited, when data is fixed but prior draws are unlimited, and in the commonplace fixed data and parameters setting. Our investigations reveal that the highest performing models are distant from the competitors (NRE-A, NRE-B) in hyperparameter space. We make a recommendation for hyperparameters distinct from the previous models. We suggest two bounds on the mutual information as performance metrics for simulation-based inference methods, without the need for posterior samples, and provide experimental results. This version corrects a minor implementation error in $γ$, improving results.

Arnaud Delaunoy, Benjamin Kurt Miller, Patrick Forré et al.

Conservative inference is a major concern in simulation-based inference. It has been shown that commonly used algorithms can produce overconfident posterior approximations. Balancing has empirically proven to be an effective way to mitigate this issue. However, its application remains limited to neural ratio estimation. In this work, we extend balancing to any algorithm that provides a posterior density. In particular, we introduce a balanced version of both neural posterior estimation and contrastive neural ratio estimation. We show empirically that the balanced versions tend to produce conservative posterior approximations on a wide variety of benchmarks. In addition, we provide an alternative interpretation of the balancing condition in terms of the $χ^2$ divergence.

Teodora Pandeva, Patrick Forré, Aaditya Ramdas et al.

We propose a general framework for constructing powerful, sequential hypothesis tests for a large class of nonparametric testing problems. The null hypothesis for these problems is defined in an abstract form using the action of two known operators on the data distribution. This abstraction allows for a unified treatment of several classical tasks, such as two-sample testing, independence testing, and conditional-independence testing, as well as modern problems, such as testing for adversarial robustness of machine learning (ML) models. Our proposed framework has the following advantages over classical batch tests: 1) it continuously monitors online data streams and efficiently aggregates evidence against the null, 2) it provides tight control over the type I error without the need for multiple testing correction, 3) it adapts the sample size requirement to the unknown hardness of the problem. We develop a principled approach of leveraging the representation capability of ML models within the testing-by-betting framework, a game-theoretic approach for designing sequential tests. Empirical results on synthetic and real-world datasets demonstrate that tests instantiated using our general framework are competitive against specialized baselines on several tasks.

Jim Boelrijk, Bernd Ensing, Patrick Forré

Optimizing multiple competing objectives is a common problem across science and industry. The inherent inextricable trade-off between those objectives leads one to the task of exploring their Pareto front. A meaningful quantity for the purpose of the latter is the hypervolume indicator, which is used in Bayesian Optimization (BO) and Evolutionary Algorithms (EAs). However, the computational complexity for the calculation of the hypervolume scales unfavorably with increasing number of objectives and data points, which restricts its use in those common multi-objective optimization frameworks. To overcome these restrictions we propose to approximate the hypervolume function with a deep neural network, which we call DeepHV. For better sample efficiency and generalization, we exploit the fact that the hypervolume is scale-equivariant in each of the objectives as well as permutation invariant w.r.t. both the objectives and the samples, by using a deep neural network that is equivariant w.r.t. the combined group of scalings and permutations. We evaluate our method against exact, and approximate hypervolume methods in terms of accuracy, computation time, and generalization. We also apply and compare our methods to state-of-the-art multi-objective BO methods and EAs on a range of synthetic benchmark test cases. The results show that our methods are promising for such multi-objective optimization tasks.

Teodora Pandeva, Martijs Jonker, Leendert Hamoen et al.

Unraveling the co-expression of genes across studies enhances the understanding of cellular processes. Inferring gene co-expression networks from transcriptome data presents many challenges, including spurious gene correlations, sample correlations, and batch effects. To address these complexities, we introduce a robust method for high-dimensional graph inference from multiple independent studies. We base our approach on the premise that each dataset is essentially a noisy linear mixture of gene loadings that follow a multivariate $t$-distribution with a sparse precision matrix, which is shared across studies. This allows us to show that we can identify the co-expression matrix up to a scaling factor among other model parameters. Our method employs an Expectation-Maximization procedure for parameter estimation. Empirical evaluation on synthetic and gene expression data demonstrates our method's improved ability to learn the underlying graph structure compared to baseline methods.

Teodora Pandeva, Tim Bakker, Christian A. Naesseth et al.

We introduce a powerful deep classifier two-sample test for high-dimensional data based on E-values, called E-value Classifier Two-Sample Test (E-C2ST). Our test combines ideas from existing work on split likelihood ratio tests and predictive independence tests. The resulting E-values are suitable for anytime-valid sequential two-sample tests. This feature allows for more effective use of data in constructing test statistics. Through simulations and real data applications, we empirically demonstrate that E-C2ST achieves enhanced statistical power by partitioning datasets into multiple batches beyond the conventional two-split (training and testing) approach of standard classifier two-sample tests. This strategy increases the power of the test while keeping the type I error well below the desired significance level.

Marco Federici, Patrick Forré, Ryota Tomioka et al.

Markov processes are widely used mathematical models for describing dynamic systems in various fields. However, accurately simulating large-scale systems at long time scales is computationally expensive due to the short time steps required for accurate integration. In this paper, we introduce an inference process that maps complex systems into a simplified representational space and models large jumps in time. To achieve this, we propose Time-lagged Information Bottleneck (T-IB), a principled objective rooted in information theory, which aims to capture relevant temporal features while discarding high-frequency information to simplify the simulation task and minimize the inference error. Our experiments demonstrate that T-IB learns information-optimal representations for accurately modeling the statistical properties and dynamics of the original process at a selected time lag, outperforming existing time-lagged dimensionality reduction methods.

Mircea Mironenco, Patrick Forré

Invariance and equivariance to geometrical transformations have proven to be very useful inductive biases when training (convolutional) neural network models, especially in the low-data regime. Much work has focused on the case where the symmetry group employed is compact or abelian, or both. Recent work has explored enlarging the class of transformations used to the case of Lie groups, principally through the use of their Lie algebra, as well as the group exponential and logarithm maps. The applicability of such methods is limited by the fact that depending on the group of interest $G$, the exponential map may not be surjective. Further limitations are encountered when $G$ is neither compact nor abelian. Using the structure and geometry of Lie groups and their homogeneous spaces, we present a framework by which it is possible to work with such groups primarily focusing on the groups $G = \text{GL}^{+}(n, \mathbb{R})$ and $G = \text{SL}(n, \mathbb{R})$, as well as their representation as affine transformations $\mathbb{R}^{n} \rtimes G$. Invariant integration as well as a global parametrization is realized by a decomposition into subgroups and submanifolds which can be handled individually. Under this framework, we show how convolution kernels can be parametrized to build models equivariant with respect to affine transformations. We evaluate the robustness and out-of-distribution generalisation capability of our model on the benchmark affine-invariant classification task, outperforming previous proposals.

Kaitlin Maile, Dennis G. Wilson, Patrick Forré

Incorporating equivariance to symmetry groups as a constraint during neural network training can improve performance and generalization for tasks exhibiting those symmetries, but such symmetries are often not perfectly nor explicitly present. This motivates algorithmically optimizing the architectural constraints imposed by equivariance. We propose the equivariance relaxation morphism, which preserves functionality while reparameterizing a group equivariant layer to operate with equivariance constraints on a subgroup, as well as the [G]-mixed equivariant layer, which mixes layers constrained to different groups to enable within-layer equivariance optimization. We further present evolutionary and differentiable neural architecture search (NAS) algorithms that utilize these mechanisms respectively for equivariance-aware architectural optimization. Experiments across a variety of datasets show the benefit of dynamically constrained equivariance to find effective architectures with approximate equivariance.

Marco Federici, David Ruhe, Patrick Forré

Estimating the mutual information from samples from a joint distribution is a challenging problem in both science and engineering. In this work, we realize a variational bound that generalizes both discriminative and generative approaches. Using this bound, we propose a hybrid method to mitigate their respective shortcomings. Further, we propose Predictive Quantization (PQ): a simple generative method that can be easily combined with discriminative estimators for minimal computational overhead. Our propositions yield a tighter bound on the information thanks to the reduced variance of the estimator. We test our methods on a challenging task of correlated high-dimensional Gaussian distributions and a stochastic process involving a system of free particles subjected to a fixed energy landscape. Empirical results show that hybrid methods consistently improved mutual information estimates when compared to the corresponding discriminative counterpart.

Teodora Pandeva, Patrick Forré

Independent component analysis (ICA) is a blind source separation method for linear disentanglement of independent latent sources from observed data. We investigate the special setting of noisy linear ICA where the observations are split among different views, each receiving a mixture of shared and individual sources. We prove that the corresponding linear structure is identifiable, and the source distribution can be recovered. To computationally estimate the sources, we optimize a constrained form of the joint log-likelihood of the observed data among all views. We also show empirically that our objective recovers the sources also in the case when the measurements are corrupted by noise. Furthermore, we propose a model selection procedure for recovering the number of shared sources which we verify empirically. Finally, we apply the proposed model in a challenging real-life application, where the estimated shared sources from two large transcriptome datasets (observed data) provided by two different labs (two different views) lead to recovering (shared) sources utilized for finding a plausible representation of the underlying graph structure.

Fiona Lippert, Bart Kranstauber, E. Emiel van Loon et al.

Studying animal movements is essential for effective wildlife conservation and conflict mitigation. For aerial movements, operational weather radars have become an indispensable data source in this respect. However, partial measurements, incomplete spatial coverage, and poor understanding of animal behaviours make it difficult to reconstruct complete spatio-temporal movement patterns from available radar data. We tackle this inverse problem by learning a mapping from high-dimensional radar measurements to low-dimensional latent representations using a convolutional encoder. Under the assumption that the latent system dynamics are well approximated by a locally linear Gaussian transition model, we perform efficient posterior estimation using the classical Kalman smoother. A convolutional decoder maps the inferred latent system states back to the physical space in which the known radar observation model can be applied, enabling fully unsupervised training. To encourage physical consistency, we additionally introduce a physics-informed loss term that leverages known mass conservation constraints. Our experiments on synthetic radar data show promising results in terms of reconstruction quality and data-efficiency.

Metod Jazbec, Patrick Forré, Stephan Mandt et al.

Early-exit neural networks (EENNs) enable adaptive and efficient inference by providing predictions at multiple stages during the forward pass. In safety-critical applications, these predictions are meaningful only when accompanied by reliable uncertainty estimates. A popular method for quantifying the uncertainty of predictive models is the use of prediction sets. However, we demonstrate that standard techniques such as conformal prediction and Bayesian credible sets are not suitable for EENNs. They tend to generate non-nested sets across exits, meaning that labels deemed improbable at one exit may reappear in the prediction set of a subsequent exit. To address this issue, we investigate anytime-valid confidence sequences (AVCSs), an extension of traditional confidence intervals tailored for data-streaming scenarios. These sequences are inherently nested and thus well-suited for an EENN's sequential predictions. We explore the theoretical and practical challenges of using AVCSs in EENNs and show that they indeed yield nested sets across exits. Thus our work presents a promising approach towards fast, yet still safe, predictive modeling

Benjamin Kurt Miller, Marco Federici, Christoph Weniger et al.

In Simulation-based Inference, the goal is to solve the inverse problem when the likelihood is only known implicitly. Neural Posterior Estimation commonly fits a normalized density estimator as a surrogate model for the posterior. This formulation cannot easily fit unnormalized surrogates because it optimizes the Kullback-Leibler divergence. We propose to optimize a generalized Kullback-Leibler divergence that accounts for the normalization constant in unnormalized distributions. The objective recovers Neural Posterior Estimation when the model class is normalized and unifies it with Neural Ratio Estimation, combining both into a single objective. We investigate a hybrid model that offers the best of both worlds by learning a normalized base distribution and a learned ratio. We also present benchmark results.

Fiona Lippert, Bart Kranstauber, E. Emiel van Loon et al.

Probabilistic inference in high-dimensional state-space models is computationally challenging. For many spatiotemporal systems, however, prior knowledge about the dependency structure of state variables is available. We leverage this structure to develop a computationally efficient approach to state estimation and learning in graph-structured state-space models with (partially) unknown dynamics and limited historical data. Building on recent methods that combine ideas from deep learning with principled inference in Gaussian Markov random fields (GMRF), we reformulate graph-structured state-space models as Deep GMRFs defined by simple spatial and temporal graph layers. This results in a flexible spatiotemporal prior that can be learned efficiently from a single time sequence via variational inference. Under linear Gaussian assumptions, we retain a closed-form posterior, which can be sampled efficiently using the conjugate gradient method, scaling favourably compared to classical Kalman filter based approaches

Fiona Lippert, Bart Kranstauber, Patrick Forré et al.

Hybrid modeling aims to augment traditional theory-driven models with machine learning components that learn unknown parameters, sub-models or correction terms from data. In this work, we build on FluxRGNN, a recently developed hybrid model of continental-scale bird migration, which combines a movement model inspired by fluid dynamics with recurrent neural networks that capture the complex decision-making processes of birds. While FluxRGNN has been shown to successfully predict key migration patterns, its spatial resolution is constrained by the typically sparse observations obtained from weather radars. Additionally, its trainable components lack explicit incentives to adequately predict take-off and landing events. Both aspects limit our ability to interpret model results ecologically. To address this, we propose two major modifications that allow for more detailed predictions on any desired tessellation while providing control over the interpretability of model components. In experiments on the U.S. weather radar network, the enhanced model effectively leverages the underlying movement model, resulting in strong extrapolation capabilities to unobserved locations.

Miriam Rateike, Isabel Valera, Patrick Forré

Neglecting the effect that decisions have on individuals (and thus, on the underlying data distribution) when designing algorithmic decision-making policies may increase inequalities and unfairness in the long term - even if fairness considerations were taken in the policy design process. In this paper, we propose a novel framework for achieving long-term group fairness in dynamical systems, in which current decisions may affect an individual's features in the next step, and thus, future decisions. Specifically, our framework allows us to identify a time-independent policy that converges, if deployed, to the targeted fair stationary state of the system in the long term, independently of the initial data distribution. We model the system dynamics with a time-homogeneous Markov chain and optimize the policy leveraging the Markov chain convergence theorem to ensure unique convergence. We provide examples of different targeted fair states of the system, encompassing a range of long-term goals for society and policymakers. Furthermore, we show how our approach facilitates the evaluation of different long-term targets by examining their impact on the group-conditional population distribution in the long term and how it evolves until convergence.

Cong Liu, David Ruhe, Patrick Forré

Most current deep learning models equivariant to $O(n)$ or $SO(n)$ either consider mostly scalar information such as distances and angles or have a very high computational complexity. In this work, we test a few novel message passing graph neural networks (GNNs) based on Clifford multivectors, structured similarly to other prevalent equivariant models in geometric deep learning. Our approach leverages efficient invariant scalar features while simultaneously performing expressive learning on multivector representations, particularly through the use of the equivariant geometric product operator. By integrating these elements, our methods outperform established efficient baseline models on an N-Body simulation task and protein denoising task while maintaining a high efficiency. In particular, we push the state-of-the-art error on the N-body dataset to 0.0035 (averaged over 3 runs); an 8% improvement over recent methods. Our implementation is available on Github.

Andrei Apostol, Maarten Stol, Patrick Forré

Modern neural networks, although achieving state-of-the-art results on many tasks, tend to have a large number of parameters, which increases training time and resource usage. This problem can be alleviated by pruning. Existing methods, however, often require extensive parameter tuning or multiple cycles of pruning and retraining to convergence in order to obtain a favorable accuracy-sparsity trade-off. To address these issues, we propose a novel pruning method which uses the oscillations around $0$ (i.e. sign flips) that a weight has undergone during training in order to determine its saliency. Our method can perform pruning before the network has converged, requires little tuning effort due to having good default values for its hyperparameters, and can directly target the level of sparsity desired by the user. Our experiments, performed on a variety of object classification architectures, show that it is competitive with existing methods and achieves state-of-the-art performance for levels of sparsity of $99.6\%$ and above for most of the architectures tested. For reproducibility, we release our code publicly at https://github.com/AndreiXYZ/flipout.

Maksim Zhdanov, David Ruhe, Maurice Weiler et al.

We present Clifford-Steerable Convolutional Neural Networks (CS-CNNs), a novel class of $\mathrm{E}(p, q)$-equivariant CNNs. CS-CNNs process multivector fields on pseudo-Euclidean spaces $\mathbb{R}^{p,q}$. They cover, for instance, $\mathrm{E}(3)$-equivariance on $\mathbb{R}^3$ and Poincaré-equivariance on Minkowski spacetime $\mathbb{R}^{1,3}$. Our approach is based on an implicit parametrization of $\mathrm{O}(p,q)$-steerable kernels via Clifford group equivariant neural networks. We significantly and consistently outperform baseline methods on fluid dynamics as well as relativistic electrodynamics forecasting tasks.

Cong Liu, David Ruhe, Floor Eijkelboom et al.

We introduce Clifford Group Equivariant Simplicial Message Passing Networks, a method for steerable E(n)-equivariant message passing on simplicial complexes. Our method integrates the expressivity of Clifford group-equivariant layers with simplicial message passing, which is topologically more intricate than regular graph message passing. Clifford algebras include higher-order objects such as bivectors and trivectors, which express geometric features (e.g., areas, volumes) derived from vectors. Using this knowledge, we represent simplex features through geometric products of their vertices. To achieve efficient simplicial message passing, we share the parameters of the message network across different dimensions. Additionally, we restrict the final message to an aggregation of the incoming messages from different dimensions, leading to what we term shared simplicial message passing. Experimental results show that our method is able to outperform both equivariant and simplicial graph neural networks on a variety of geometric tasks.

Maksim Zhdanov, Nabil Iqbal, Erik Bekkers et al.

Conformal symmetries, i.e.\ coordinate transformations that preserve angles, play a key role in many fields, including physics, mathematics, computer vision and (geometric) machine learning. Here we build a neural network that is equivariant under general conformal transformations. To achieve this, we lift data from flat Euclidean space to Anti de Sitter (AdS) space. This allows us to exploit a known correspondence between conformal transformations of flat space and isometric transformations on the AdS space. We then build upon the fact that such isometric transformations have been extensively studied on general geometries in the geometric deep learning literature. We employ message-passing layers conditioned on the proper distance, yielding a computationally efficient framework. We validate our model on tasks from computer vision and statistical physics, demonstrating strong performance, improved generalization capacities, and the ability to extract conformal data such as scaling dimensions from the trained network.

Cong Liu, Sharvaree Vadgama, David Ruhe et al.

This paper explores leveraging the Clifford algebra's expressive power for $\E(n)$-equivariant diffusion models. We utilize the geometric products between Clifford multivectors and the rich geometric information encoded in Clifford subspaces in \emph{Clifford Diffusion Models} (CDMs). We extend the diffusion process beyond just Clifford one-vectors to incorporate all higher-grade multivector subspaces. The data is embedded in grade-$k$ subspaces, allowing us to apply latent diffusion across complete multivectors. This enables CDMs to capture the joint distribution across different subspaces of the algebra, incorporating richer geometric information through higher-order features. We provide empirical results for unconditional molecular generation on the QM9 dataset, showing that CDMs provide a promising avenue for generative modeling.

Patrick Forré, Abel Jansma

We generalize the concept of Möbius inversion and Shapley values to directed acyclic multigraphs and weighted versions thereof. We further allow value functions (games) and thus their Möbius transforms (synergy function) and Shapley values to have values in any abelian group that is a module over a ring that contains the graph weights, e.g. vector-valued functions. To achieve this and overcome the obstruction that the classical axioms (linearity, efficiency, null player, symmetry) are not strong enough to uniquely determine Shapley values in this more general setting, we analyze Shapley values from two novel points of view: 1) We introduce projection operators that allow us to interpret Shapley values as the recursive projection and re-attribution of higher-order synergies to lower-order ones; 2) we propose a strengthening of the null player axiom and a localized symmetry axiom, namely the weak elements and flat hierarchy axioms. The former allows us to remove coalitions with vanishing synergy while preserving the rest of the hierarchical structure. The latter treats player-coalition bonds uniformly in the corner case of hierarchically flat graphs. Together with linearity these axioms already imply a unique explicit formula for the Shapley values, as well as classical properties like efficiency, null player, symmetry, and novel ones like the projection property. This whole framework then specializes to finite inclusion algebras, lattices, partial orders and mereologies, and also recovers certain previously known cases as corner cases, and presents others from a new perspective. The admission of general weighted directed acyclic multigraph structured hierarchies and vector-valued functions and Shapley values opens up the possibility for new analytic tools and application areas, like machine learning, language processing, explainable artificial intelligence, and many more.

Mohan Guo, Cong Liu, Patrick Forré

Recent methods for molecular generation face a trade-off: they either enforce strict equivariance with costly architectures or relax it to gain scalability and flexibility. We propose a frame-based diffusion paradigm that achieves deterministic E(3)-equivariance while decoupling symmetry handling from the backbone. Building on this paradigm, we investigate three variants: Global Frame Diffusion (GFD), which assigns a shared molecular frame; Local Frame Diffusion (LFD), which constructs node-specific frames and benefits from additional alignment constraints; and Invariant Frame Diffusion (IFD), which relies on pre-canonicalized invariant representations. To enhance expressivity, we further utilize EdgeDiT, a Diffusion Transformer with edge-aware attention. On the QM9 dataset, GFD with EdgeDiT achieves state-of-the-art performance, with a test NLL of -137.97 at standard scale and -141.85 at double scale, alongside atom stability of 98.98%, and molecular stability of 90.51%. These results surpass all equivariant baselines while maintaining high validity and uniqueness and nearly 2x faster sampling compared to EDM. Altogether, our study establishes frame-based diffusion as a scalable, flexible, and physically grounded paradigm for molecular generation, highlighting the critical role of global structure preservation.

Leon Lang, Patrick Forré

As AI systems advance beyond human capabilities, scalable oversight becomes critical: how can we supervise AI that exceeds our abilities? A key challenge is that human evaluators may form incorrect beliefs about AI behavior in complex tasks, leading to unreliable feedback and poor value inference. To address this, we propose modeling evaluators' beliefs to interpret their feedback more reliably. We formalize human belief models, analyze their theoretical role in value learning, and characterize when ambiguity remains. To reduce reliance on precise belief models, we introduce "belief model covering" as a relaxation. This motivates our preliminary proposal to use the internal representations of adapted foundation models to mimic human evaluators' beliefs. These representations could be used to learn correct values from human feedback even when evaluators misunderstand the AI's behavior. Our work suggests that modeling human beliefs can improve value learning and outlines practical research directions for implementing this approach to scalable oversight.

Lukas Fluri, Leon Lang, Alessandro Abate et al.

In reinforcement learning, specifying reward functions that capture the intended task can be very challenging. Reward learning aims to address this issue by learning the reward function. However, a learned reward model may have a low error on the data distribution, and yet subsequently produce a policy with large regret. We say that such a reward model has an error-regret mismatch. The main source of an error-regret mismatch is the distributional shift that commonly occurs during policy optimization. In this paper, we mathematically show that a sufficiently low expected test error of the reward model guarantees low worst-case regret, but that for any fixed expected test error, there exist realistic data distributions that allow for error-regret mismatch to occur. We then show that similar problems persist even when using policy regularization techniques, commonly employed in methods such as RLHF. We hope our results stimulate the theoretical and empirical study of improved methods to learn reward models, and better ways to measure their quality reliably.

David Ruhe, Johannes Brandstetter, Patrick Forré

We introduce Clifford Group Equivariant Neural Networks: a novel approach for constructing $\mathrm{O}(n)$- and $\mathrm{E}(n)$-equivariant models. We identify and study the $\textit{Clifford group}$, a subgroup inside the Clifford algebra tailored to achieve several favorable properties. Primarily, the group's action forms an orthogonal automorphism that extends beyond the typical vector space to the entire Clifford algebra while respecting the multivector grading. This leads to several non-equivalent subrepresentations corresponding to the multivector decomposition. Furthermore, we prove that the action respects not just the vector space structure of the Clifford algebra but also its multiplicative structure, i.e., the geometric product. These findings imply that every polynomial in multivectors, An advantage worth mentioning is that we obtain expressive layers that can elegantly generalize to inner-product spaces of any dimension. We demonstrate, notably from a single core implementation, state-of-the-art performance on several distinct tasks, including a three-dimensional $n$-body experiment, a four-dimensional Lorentz-equivariant high-energy physics experiment, and a five-dimensional convex hull experiment.

David Ruhe, Patrick Forré

We perform approximate inference in state-space models with nonlinear state transitions. Without parameterizing a generative model, we apply Bayesian update formulas using a local linearity approximation parameterized by neural networks. This comes accompanied by a maximum likelihood objective that requires no supervision via uncorrupt observations or ground truth latent states. The optimization backpropagates through a recursion similar to the classical Kalman filter and smoother. Additionally, using an approximate conditional independence, we can perform smoothing without having to parameterize a separate model. In scientific applications, domain knowledge can give a linear approximation of the latent transition maps, which we can easily incorporate into our model. Usage of such domain knowledge is reflected in excellent results (despite our model's simplicity) on the chaotic Lorenz system compared to fully supervised and variational inference methods. Finally, we show competitive results on an audio denoising experiment.

Benjamin Kurt Miller, Alex Cole, Patrick Forré et al.

Parametric stochastic simulators are ubiquitous in science, often featuring high-dimensional input parameters and/or an intractable likelihood. Performing Bayesian parameter inference in this context can be challenging. We present a neural simulation-based inference algorithm which simultaneously offers simulation efficiency and fast empirical posterior testability, which is unique among modern algorithms. Our approach is simulation efficient by simultaneously estimating low-dimensional marginal posteriors instead of the joint posterior and by proposing simulations targeted to an observation of interest via a prior suitably truncated by an indicator function. Furthermore, by estimating a locally amortized posterior our algorithm enables efficient empirical tests of the robustness of the inference results. Since scientists cannot access the ground truth, these tests are necessary for trusting inference in real-world applications. We perform experiments on a marginalized version of the simulation-based inference benchmark and two complex and narrow posteriors, highlighting the simulator efficiency of our algorithm as well as the quality of the estimated marginal posteriors.

Maurice Weiler, Patrick Forré, Erik Verlinde et al.

Motivated by the vast success of deep convolutional networks, there is a great interest in generalizing convolutions to non-Euclidean manifolds. A major complication in comparison to flat spaces is that it is unclear in which alignment a convolution kernel should be applied on a manifold. The underlying reason for this ambiguity is that general manifolds do not come with a canonical choice of reference frames (gauge). Kernels and features therefore have to be expressed relative to arbitrary coordinates. We argue that the particular choice of coordinatization should not affect a network's inference -- it should be coordinate independent. A simultaneous demand for coordinate independence and weight sharing is shown to result in a requirement on the network to be equivariant under local gauge transformations (changes of local reference frames). The ambiguity of reference frames depends thereby on the G-structure of the manifold, such that the necessary level of gauge equivariance is prescribed by the corresponding structure group G. Coordinate independent convolutions are proven to be equivariant w.r.t. those isometries that are symmetries of the G-structure. The resulting theory is formulated in a coordinate free fashion in terms of fiber bundles. To exemplify the design of coordinate independent convolutions, we implement a convolutional network on the Möbius strip. The generality of our differential geometric formulation of convolutional networks is demonstrated by an extensive literature review which explains a large number of Euclidean CNNs, spherical CNNs and CNNs on general surfaces as specific instances of coordinate independent convolutions.

Marco Federici, Ryota Tomioka, Patrick Forré

Safely deploying machine learning models to the real world is often a challenging process. Models trained with data obtained from a specific geographic location tend to fail when queried with data obtained elsewhere, agents trained in a simulation can struggle to adapt when deployed in the real world or novel environments, and neural networks that are fit to a subset of the population might carry some selection bias into their decision process. In this work, we describe the problem of data shift from a novel information-theoretic perspective by (i) identifying and describing the different sources of error, (ii) comparing some of the most promising objectives explored in the recent domain generalization, and fair classification literature. From our theoretical analysis and empirical evaluation, we conclude that the model selection procedure needs to be guided by careful considerations regarding the observed data, the factors used for correction, and the structure of the data-generating process.

Patrick Forré

We develope the framework of transitional conditional independence. For this we introduce transition probability spaces and transitional random variables. These constructions will generalize, strengthen and unify previous notions of (conditional) random variables and non-stochastic variables, (extended) stochastic conditional independence and some form of functional conditional independence. Transitional conditional independence is asymmetric in general and it will be shown that it satisfies all desired relevance relations in terms of left and right versions of the separoid rules, except symmetry, on standard, analytic and universal measurable spaces. As a preparation we prove a disintegration theorem for transition probabilities, i.e. the existence and essential uniqueness of (regular) conditional Markov kernels, on those spaces. Transitional conditional independence will be able to express classical statistical concepts like sufficiency, adequacy and ancillarity. As an application, we will then show how transitional conditional independence can be used to prove a directed global Markov property for causal graphical models that allow for non-stochastic input variables in strong generality. This will then also allow us to show the main rules of causal/do-calculus, relating observational and interventional distributions, in such measure theoretic generality.

Maximilian Ilse, Patrick Forré, Max Welling et al.

Unobserved confounding is one of the main challenges when estimating causal effects. We propose a causal reduction method that, given a causal model, replaces an arbitrary number of possibly high-dimensional latent confounders with a single latent confounder that takes values in the same space as the treatment variable, without changing the observational and interventional distributions the causal model entails. This allows us to estimate the causal effect in a principled way from combined data without relying on the common but often unrealistic assumption that all confounders have been observed. We apply our causal reduction in three different settings. In the first setting, we assume the treatment and outcome to be discrete. The causal reduction then implies bounds between the observational and interventional distributions that can be exploited for estimation purposes. In certain cases with highly unbalanced observational samples, the accuracy of the causal effect estimate can be improved by incorporating observational data. Second, for continuous variables and assuming a linear-Gaussian model, we derive equality constraints for the parameters of the observational and interventional distributions. Third, for the general continuous setting (possibly nonlinear and non-Gaussian), we parameterize the reduced causal model using normalizing flows, a flexible class of easily invertible nonlinear transformations. We perform a series of experiments on synthetic data and find that in several cases the number of interventional samples can be reduced when adding observational training samples without sacrificing accuracy.

Emiel Hoogeboom, Didrik Nielsen, Priyank Jaini et al.

Generative flows and diffusion models have been predominantly trained on ordinal data, for example natural images. This paper introduces two extensions of flows and diffusion for categorical data such as language or image segmentation: Argmax Flows and Multinomial Diffusion. Argmax Flows are defined by a composition of a continuous distribution (such as a normalizing flow), and an argmax function. To optimize this model, we learn a probabilistic inverse for the argmax that lifts the categorical data to a continuous space. Multinomial Diffusion gradually adds categorical noise in a diffusion process, for which the generative denoising process is learned. We demonstrate that our method outperforms existing dequantization approaches on text modelling and modelling on image segmentation maps in log-likelihood.

T. Anderson Keller, Jorn W. T. Peters, Priyank Jaini et al.

Efficient gradient computation of the Jacobian determinant term is a core problem in many machine learning settings, and especially so in the normalizing flow framework. Most proposed flow models therefore either restrict to a function class with easy evaluation of the Jacobian determinant, or an efficient estimator thereof. However, these restrictions limit the performance of such density models, frequently requiring significant depth to reach desired performance levels. In this work, we propose Self Normalizing Flows, a flexible framework for training normalizing flows by replacing expensive terms in the gradient by learned approximate inverses at each layer. This reduces the computational complexity of each layer's exact update from $\mathcal{O}(D^3)$ to $\mathcal{O}(D^2)$, allowing for the training of flow architectures which were otherwise computationally infeasible, while also providing efficient sampling. We show experimentally that such models are remarkably stable and optimize to similar data likelihood values as their exact gradient counterparts, while training more quickly and surpassing the performance of functionally constrained counterparts.

Rik Helwegen, Christos Louizos, Patrick Forré

The importance of algorithmic fairness grows with the increasing impact machine learning has on people's lives. Recent work on fairness metrics shows the need for causal reasoning in fairness constraints. In this work, a practical method named FairTrade is proposed for creating flexible prediction models which integrate fairness constraints on sensitive causal paths. The method uses recent advances in variational inference in order to account for unobserved confounders. Further, a method outline is proposed which uses the causal mechanism estimates to audit black box models. Experiments are conducted on simulated data and on a real dataset in the context of detecting unlawful social welfare. This research aims to contribute to machine learning techniques which honour our ethical and legal boundaries.

Luca Falorsi, Patrick Forré

Normalizing flows are a powerful technique for obtaining reparameterizable samples from complex multimodal distributions. Unfortunately current approaches fall short when the underlying space has a non trivial topology, and are only available for the most basic geometries. Recently normalizing flows in Euclidean space based on Neural ODEs show great promise, yet suffer the same limitations. Using ideas from differential geometry and geometric control theory, we describe how neural ODEs can be extended to smooth manifolds. We show how vector fields provide a general framework for parameterizing a flexible class of invertible mapping on these spaces and we illustrate how gradient based learning can be performed. As a result we define a general methodology for building normalizing flows on manifolds.

Stijn Verdenius, Maarten Stol, Patrick Forré

With the introduction of SNIP [arXiv:1810.02340v2], it has been demonstrated that modern neural networks can effectively be pruned before training. Yet, its sensitivity criterion has since been criticized for not propagating training signal properly or even disconnecting layers. As a remedy, GraSP [arXiv:2002.07376v1] was introduced, compromising on simplicity. However, in this work we show that by applying the sensitivity criterion iteratively in smaller steps - still before training - we can improve its performance without difficult implementation. As such, we introduce 'SNIP-it'. We then demonstrate how it can be applied for both structured and unstructured pruning, before and/or during training, therewith achieving state-of-the-art sparsity-performance trade-offs. That is, while already providing the computational benefits of pruning in the training process from the start. Furthermore, we evaluate our methods on robustness to overfitting, disconnection and adversarial attacks as well.

Maximilian Ilse, Jakub M. Tomczak, Patrick Forré

Machine learning models trained with purely observational data and the principle of empirical risk minimization \citep{vapnik_principles_1992} can fail to generalize to unseen domains. In this paper, we focus on the case where the problem arises through spurious correlation between the observed domains and the actual task labels. We find that many domain generalization methods do not explicitly take this spurious correlation into account. Instead, especially in more application-oriented research areas like medical imaging or robotics, data augmentation techniques that are based on heuristics are used to learn domain invariant features. To bridge the gap between theory and practice, we develop a causal perspective on the problem of domain generalization. We argue that causal concepts can be used to explain the success of data augmentation by describing how they can weaken the spurious correlation between the observed domains and the task labels. We demonstrate that data augmentation can serve as a tool for simulating interventional data. We use these theoretical insights to derive a simple algorithm that is able to select data augmentation techniques that will lead to better domain generalization.

Marco Federici, Anjan Dutta, Patrick Forré et al.

The information bottleneck principle provides an information-theoretic method for representation learning, by training an encoder to retain all information which is relevant for predicting the label while minimizing the amount of other, excess information in the representation. The original formulation, however, requires labeled data to identify the superfluous information. In this work, we extend this ability to the multi-view unsupervised setting, where two views of the same underlying entity are provided but the label is unknown. This enables us to identify superfluous information as that not shared by both views. A theoretical analysis leads to the definition of a new multi-view model that produces state-of-the-art results on the Sketchy dataset and label-limited versions of the MIR-Flickr dataset. We also extend our theory to the single-view setting by taking advantage of standard data augmentation techniques, empirically showing better generalization capabilities when compared to common unsupervised approaches for representation learning.

Luca Falorsi, Pim de Haan, Tim R. Davidson et al.

Reparameterizable densities are an important way to learn probability distributions in a deep learning setting. For many distributions it is possible to create low-variance gradient estimators by utilizing a `reparameterization trick'. Due to the absence of a general reparameterization trick, much research has recently been devoted to extend the number of reparameterizable distributional families. Unfortunately, this research has primarily focused on distributions defined in Euclidean space, ruling out the usage of one of the most influential class of spaces with non-trivial topologies: Lie groups. In this work we define a general framework to create reparameterizable densities on arbitrary Lie groups, and provide a detailed practitioners guide to further the ease of usage. We demonstrate how to create complex and multimodal distributions on the well known oriented group of 3D rotations, $\operatorname{SO}(3)$, using normalizing flows. Our experiments on applying such distributions in a Bayesian setting for pose estimation on objects with discrete and continuous symmetries, showcase their necessity in achieving realistic uncertainty estimates.

Patrick Forré, Joris M. Mooij

We prove the main rules of causal calculus (also called do-calculus) for i/o structural causal models (ioSCMs), a generalization of a recently proposed general class of non-/linear structural causal models that allow for cycles, latent confounders and arbitrary probability distributions. We also generalize adjustment criteria and formulas from the acyclic setting to the general one (i.e. ioSCMs). Such criteria then allow to estimate (conditional) causal effects from observational data that was (partially) gathered under selection bias and cycles. This generalizes the backdoor criterion, the selection-backdoor criterion and extensions of these to arbitrary ioSCMs. Together, our results thus enable causal reasoning in the presence of cycles, latent confounders and selection bias. Finally, we extend the ID algorithm for the identification of causal effects to ioSCMs.

Giorgio Patrini, Rianne van den Berg, Patrick Forré et al.

Optimal transport offers an alternative to maximum likelihood for learning generative autoencoding models. We show that minimizing the p-Wasserstein distance between the generator and the true data distribution is equivalent to the unconstrained min-min optimization of the p-Wasserstein distance between the encoder aggregated posterior and the prior in latent space, plus a reconstruction error. We also identify the role of its trade-off hyperparameter as the capacity of the generator: its Lipschitz constant. Moreover, we prove that optimizing the encoder over any class of universal approximators, such as deterministic neural networks, is enough to come arbitrarily close to the optimum. We therefore advertise this framework, which holds for any metric space and prior, as a sweet-spot of current generative autoencoding objectives. We then introduce the Sinkhorn auto-encoder (SAE), which approximates and minimizes the p-Wasserstein distance in latent space via backprogation through the Sinkhorn algorithm. SAE directly works on samples, i.e. it models the aggregated posterior as an implicit distribution, with no need for a reparameterization trick for gradients estimations. SAE is thus able to work with different metric spaces and priors with minimal adaptations. We demonstrate the flexibility of SAE on latent spaces with different geometries and priors and compare with other methods on benchmark data sets.

Luca Falorsi, Pim de Haan, Tim R. Davidson et al.

The manifold hypothesis states that many kinds of high-dimensional data are concentrated near a low-dimensional manifold. If the topology of this data manifold is non-trivial, a continuous encoder network cannot embed it in a one-to-one manner without creating holes of low density in the latent space. This is at odds with the Gaussian prior assumption typically made in Variational Auto-Encoders (VAEs), because the density of a Gaussian concentrates near a blob-like manifold. In this paper we investigate the use of manifold-valued latent variables. Specifically, we focus on the important case of continuously differentiable symmetry groups (Lie groups), such as the group of 3D rotations $\operatorname{SO}(3)$. We show how a VAE with $\operatorname{SO}(3)$-valued latent variables can be constructed, by extending the reparameterization trick to compact connected Lie groups. Our experiments show that choosing manifold-valued latent variables that match the topology of the latent data manifold, is crucial to preserve the topological structure and learn a well-behaved latent space.

Patrick Forré, Joris M. Mooij

We address the problem of causal discovery from data, making use of the recently proposed causal modeling framework of modular structural causal models (mSCM) to handle cycles, latent confounders and non-linearities. We introduce σ-connection graphs (σ-CG), a new class of mixed graphs (containing undirected, bidirected and directed edges) with additional structure, and extend the concept of σ-separation, the appropriate generalization of the well-known notion of d-separation in this setting, to apply to σ-CGs. We prove the closedness of σ-separation under marginalisation and conditioning and exploit this to implement a test of σ-separation on a σ-CG. This then leads us to the first causal discovery algorithm that can handle non-linear functional relations, latent confounders, cyclic causal relationships, and data from different (stochastic) perfect interventions. As a proof of concept, we show on synthetic data how well the algorithm recovers features of the causal graph of modular structural causal models.

Patrick Forré, Joris M. Mooij

We investigate probabilistic graphical models that allow for both cycles and latent variables. For this we introduce directed graphs with hyperedges (HEDGes), generalizing and combining both marginalized directed acyclic graphs (mDAGs) that can model latent (dependent) variables, and directed mixed graphs (DMGs) that can model cycles. We define and analyse several different Markov properties that relate the graphical structure of a HEDG with a probability distribution on a corresponding product space over the set of nodes, for example factorization properties, structural equations properties, ordered/local/global Markov properties, and marginal versions of these. The various Markov properties for HEDGes are in general not equivalent to each other when cycles or hyperedges are present, in contrast with the simpler case of directed acyclic graphical (DAG) models (also known as Bayesian networks). We show how the Markov properties for HEDGes - and thus the corresponding graphical Markov models - are logically related to each other.

Stephan Bongers, Patrick Forré, Jonas Peters et al.

Structural causal models (SCMs), also known as (nonparametric) structural equation models (SEMs), are widely used for causal modeling purposes. In particular, acyclic SCMs, also known as recursive SEMs, form a well-studied subclass of SCMs that generalize causal Bayesian networks to allow for latent confounders. In this paper, we investigate SCMs in a more general setting, allowing for the presence of both latent confounders and cycles. We show that in the presence of cycles, many of the convenient properties of acyclic SCMs do not hold in general: they do not always have a solution; they do not always induce unique observational, interventional and counterfactual distributions; a marginalization does not always exist, and if it exists the marginal model does not always respect the latent projection; they do not always satisfy a Markov property; and their graphs are not always consistent with their causal semantics. We prove that for SCMs in general each of these properties does hold under certain solvability conditions. Our work generalizes results for SCMs with cycles that were only known for certain special cases so far. We introduce the class of simple SCMs that extends the class of acyclic SCMs to the cyclic setting, while preserving many of the convenient properties of acyclic SCMs. With this paper we aim to provide the foundations for a general theory of statistical causal modeling with SCMs.